The affine SL2-surface X = Spec(D^{Z2}) embeds in A^4 but not A^3, its SL2-action never extends to any A^4 embedding, and it is non-cancellative with automorphism group PSL2(k) *_H T.
Daigle, On locally nilpotent derivations of k[x1, x 2, y ]/ (φ (y) − x1x2), J
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A note on smooth $SL_2$-surfaces
The affine SL2-surface X = Spec(D^{Z2}) embeds in A^4 but not A^3, its SL2-action never extends to any A^4 embedding, and it is non-cancellative with automorphism group PSL2(k) *_H T.